A�ne long term yield curves : an application of the

Ramsey rule with progressive utility

El Karoui Nicole, ∗ Mrad Mohamed † Hillairet Caroline ‡§

February 21, 2014

Abstract

The purpose of this paper relies on the study of long term a�ne yield curves modeling.

It is inspired by the Ramsey rule of the economic literature, that links discount rate and

marginal utility of aggregate optimal consumption. For such a long maturity modelization,

the possibility of adjusting preferences to new economic information is crucial, justifying

the use of progressive utility. This paper studies, in a framework with a�ne factors, the

yield curve given from the Ramsey rule. It �rst characterizes consistent progressive utility

of investment and consumption, given the optimal wealth and consumption processes. A

special attention is paid to utilities associated with linear optimal processes with respect

to their initial conditions, which is for example the case of power progressive utilities.

Those utilities are the basis point to construct other progressive utilities generating non

linear optimal processes but leading yet to still tractable computations. This is of partic-

ular interest to study the impact of initial wealth on yield curves.

Keywords: Progressive utility with consumption, market consistency, portfolio opti-

mization, Ramsey rule, a�ne yields curves.

Introduction

This paper focuses on the modelization of long term a�ne yield curves. For the �nancing

of ecological project, for the pricing of longevity-linked securities or any other investment

with long term impact, modeling long term interest rates is crucial. The answer cannot be

∗LPMA, UMR CNRS 6632, Université Pierre et Marie Curie, CMAP, UMR CNRS 7641, École

Polytechnique†LAGA, UMR CNRS 7539, Université Paris 13‡CMAP, UMR CNRS 7641, École Polytechnique,§With the �nancial support of the "Chaire Risque Financier of the Fondation du Risque",

1

�nd in �nancial market since for longer maturities, the bond market is highly illiquid and

standard �nancial interest rates models cannot be easily extended. Nevertheless, an abun-

dant literature on the economic aspects of long-term policy-making has been developed.

The Ramsey rule, introduced by Ramsey in his seminal work [26] and further discussed by

numerous economists such as Gollier [4, 7, 11, 6, 8, 10, 5, 9] and Weitzman [28, 29], is the

reference equation to compute discount rate, that allows to evaluate the future value of an

investment by giving a current equivalent value. The Ramsey rule links the discount rate

with the marginal utility of aggregate consumption at the economic equilibrium. Even

if this rule is very simple, there is no consensus among economists about the parameters

that should be considered, leading to very di�erent discount rates. But economists agree

on the necessity of a sequential decision scheme that allows to revise the �rst decisions

in the light of new knowledge and direct experiences: the utility criterion must be adap-

tative and adjusted to the information �ow. In the classical optimization point of view,

this adaptative criteria is called consistency. In that sense, market-consistent progressive

utilities, studied in El Karoui and Mrad [15, 14, 13], are the appropriate tools to study

long term yield curves.

Indeed, in a dynamic and stochastic environment, the classical notion of utility func-

tion is not �exible enough to help us to make good choices in the long run. M. Musiela

and T. Zariphopoulou (2003-2008 [21, 22, 20, 19]) were the �rst to suggest to use instead

of the classical criterion the concept of progressive dynamic utility, consistent with re-

spect to a given investment universe in a sense speci�ed in Section 1. The concept of

progressive utility gives an adaptative way to model possible changes over the time of

individual preferences of an agent. In continuation of the recent works of El Karoui and

Mrad [15, 14, 13], and motived by the Ramsey rule (in which the consumption rate is a key

process), [16] extends the notion of market-consistent progressive utility to the case with

consumption: the agent invest in a �nancial market and consumes a part of her wealth

at each instant. As an example, backward classical value function is a progressive utility,

the way the classical optimization problem is posed is very di�erent from the progressive

utility problem. In the classical approach, the optimal processes are computed through a

backward analysis, emphasizing their dependency to the horizon of the optimization prob-

lem, while the forward point of view makes clear the monotony of the optimal processes

to their initial conditions. A special attention is paid to progressive utilities generating

linear optimal processes with respect to their initial conditions, which is for example the

case of power progressive utilities.

As the zero-coupon bond market is highly illiquid for long maturity, it is relevant, for

small trades, to give utility indi�erence price (also called Davis price) for zero coupon,

using progressive utility with consumption. We study then the dynamics of the marginal

utility yield curve, in the framework of progressive and backward power utilities (since

power utilities are the most commonly used in the economic literature) and in a model with

a�ne factors, since this model has the advantage to lead to tractable computations while

2

allowing for more stochasticity than the log normal model studied in [16]. Nevertheless,

using power utilities implies that the impact of the initial economic wealth is avoided, since

in this case the optimal processes are linear with respect to the initial conditions. We

thus propose a way of constructing, from power utilities, progressive utilities generating

non linear optimal processes but leading yet to still tractable computations. The impact

of the initial wealth for yield curves is discussed.

The paper is organized as follows. After introducing the investment universe, Section

1 characterizes consistent progressive utility of investment and consumption, given the

optimal wealth and consumption processes. Section 2 deals with the computation of the

marginal utility yield curve, inspired by the Ramsey rule. Section 3 focuses on the yield

curve with a�ne factors, in such a setting the yield curve does not depend on the initial

wealth of the economy. Section 4 provides then a modelization for yield curves dynamics

that are non-linear to initial conditions.

1 Progressive Utility and Investment Universe

1.1 The investment universe

We consider an incomplete Itô market, equipped with a n-standard Brownian motion, W

and characterized by an adapted short rate (rt) and an adapted n-dimensional risk pre-

mium vector (ηt). All these processes are de�ned on a �ltered probability space (Ω,Ft,P)satisfying usual assumptions; they are progressively processes satisfying minimal integra-

bility assumptions, as∫ T

0 (rt + ‖ηt‖2dt)

The following short notations will be used extensively. Let R be a vector subspace of Rn.For any x ∈ Rn, xR is the orthogonal projection of the vector x onto R and x⊥ is theorthogonal projection onto R⊥.The existence of a risk premium η is a possible formulation of the absence of arbitrage

opportunity. From Equation (1.1), the minimal state price process Y 0t , whose the dynam-

ics is dY 0t = Y0t [−rtdt+ (νt − ηRt ).dWt], belongs to the convex family Y of positive Itô's

processes Yt such that (YtXκ,Ct +

∫ t0 YsCsds) is a local martingale for any admissible port-

folio. The existence of equivalent martingale measure is obtained by the assumption that

the exponential local martingale LηR

t = exp(−∫ t

0 ηRs .dWs − 12

∫ t0 ||η

Rs ||2 ds) is a uniformly

integrable martingale. Nevertheless, we are interested into the class of the so-called state

price processes Yt belonging to the family Y characterized below.

De�nition 1.2 (State price process). (i) An Itô semimartingale Yt is called a state price

process in Y if for any test process Xκ,C , κ ∈ R,(YtX

κ,Ct +

∫ t0 YsCsds) is a local martingale.

(ii) This property is equivalent to the existence of progressive process νt ∈ R⊥t , (∫ T

0 ‖νt‖2dt <

∞, a.s.) such that Y = Y ν where Y ν is the product of Y 0 (ν = 0) by the exponential localmartingale Lνt = exp

( ∫ t0 νs.dWs − 1/2

∫ t0 ||νs||

2ds), and satis�es

dY νt = Yνt [−rtdt+ (νt − ηRt ).dWt], νt ∈ R⊥t Y ν0 = y (1.2)

From now on, to stress out the dependency on the initial condition, the solution of (1.2)

with initial condition y will be denoted (Y νt (y)) and Yνt := Y

νt (1); the solution of (1.1)

with initial condition x will be denoted (Xκ,Ct (x)) and Xκ,Ct := X

κ,Ct (1).

1.2 X c-consistent Utility and Portfolio optimization with

consumption

In long term (wealth-consumption) optimization problems, it is useful to have the choice

to adapt utility criteria to deep macro-evolution of economic environment. The concept of

progressive utility is introduced in this sense. As we are interested in optimizing both the

terminal wealth and the consumption rate, we introduce two progressive utilities (U,V),

U for the terminal wealth and V for the consumption rate, often called utility system.

For sake of completeness, we start refer the reader to [15] for a detailled study.

De�nition 1.3 (Progressive Utility).

(i) A progressive utility is a C2- progressive random �eld on R∗+, U = {U(t, x); t ≥ 0, x >0}, starting from the deterministic utility function u at time 0, such that for every (t, ω),x 7→ U(ω, t, x) is a strictly concave, strictly increasing, and non negative utility function,and satisfying thee Inada conditions:

− for every (t, ω), U(t, ω, x) goes to 0 when x goes to 0− the derivative Ux(t, ω, x) (also called marginal utility) goes to ∞ when x goes to 0,− the derivative Ux(t, ω, x) goes to 0 when x goes to ∞.

4

For t = 0, the deterministic utilities U(0, .) and V (0, .) are denoted u(.) and v(.) and in

the following small letters u and v design deterministic utilities while capital letters refer

to progressive utilities.

As in statistical learning, the utility criterium is dynamically adjusted to be the best

given the market past information. So, market inputs may be viewed as a calibration

universe through the test-class X c of processes on which the utility is chosen to provide

the best satisfaction. This motivates the following de�nition of X c-consistent utility

system.

De�nition 1.4. A X c-consistent progressive utility system of investment and consump-

tion is a pair of progressive utilities U and V on Ω× [0,+∞)× R+ such that,(i) Consistency with the test-class: For any admissible wealth process Xκ,C ∈X c,

E(U(t,Xκ,Ct ) +

∫ tsV (s, Cs)ds/Fs

)≤ U(s,Xκ,Cs ), ∀s ≤ t a.s.

In other words, the process(U(t,Xκ,Ct ) +

∫ t0 V (s, Cs)ds

)is a positive supermartingale,

stopped at the �rst time of bankruptcy.

(ii) Existence of optimal strategy: For any initial wealth x > 0, there exists

an optimal strategy (κ∗, C∗) such that the associated non negative wealth process X∗ =

Xκ∗,C∗ ∈X c issued from x satis�es

(U(t,X∗t ) +

∫ t0 V (s, C

∗s )ds

)is a local martingale.

(iii) To summarize, U(t, x) is the value function of optimization problem with optimal

strategies, that is for any maturity T ≥ t

U(t, x) = ess supXκ,C∈X ct (x) E(U(T,Xκ,CT ) +

∫ TtV (s, Cs)1{Xκ,Cs (x)≥0}ds|Ft

)a.s. (1.3)

The optimal strategy (X∗, C∗) which is optimal for all these problems, independently of

the time-horizon T , is called a myopic strategy.

(iv)Strongly X c-consistency The system (U,V) is said to be strongly X c-consistent

if the optimal process X∗(x) is strictly increasing with respect to the initial condition x.

Convex analysis showed the interest to introduce the convex conjugate utilities Ũ and Ṽ

de�ned as the Fenchel-Legendre random �eld Ũ(t, y) = supc≥0,c∈Q+(U(t, c)−cy) (similarlyfor Ṽ ). Under mild regularity assumption, we have the following results (Karatzas-Shreve

[12], Rogers [27]).

Proposition 1.5 (Duality). Let (U, V ) be a pair of stochastic X c-consistent utilities with

optimal strategy (κ∗, C∗) leading to the non negative wealth process X∗ = Xκ∗,C∗. Then

the convex conjugate system (Ũ , Ṽ ) satis�es :

(i) For any admissible state price density process Y ν ∈ Y with ν ∈ R⊥,(Ũ(t, Y νt ) +∫ t

0 Ṽ (s, Yνs )ds

)is a submartingale, and there exists a unique optimal process Y ∗ := Y ν

∗

with ν∗ ∈ R⊥ such that(Ũ(t, Y ∗t ) +

∫ t0 Ṽ (s, Y

∗s )ds

)is a local martingale.

(ii) To summarize, Ũ(t, y) is the value function of optimization problem with myopic

5

optimal strategy, that is for any maturity T ≥ t

Ũ(t, y) = ess supY ν∈Yt(y) E(Ũ(T, Y νT (y)) +

∫ TtṼ (s, Y νs )(y)ds/Ft

), a.s. (1.4)

(iii) Optimal Processes characterization Under regularity assumption, �rst order

conditions imply some links between optimal processes, including their initial conditions,

Y ∗t (y) = Ux(t,X∗t (x)) = Vc(t, C

∗t (c)), y = ux(x) = vc(c) (1.5)

The optimal consumption process C∗t (c) is related to the optimal portfolio X∗t (x) by the

progressive monotonic process ζ∗t (x) de�ned by

c = ζ(x) = −ṽy(ux(x)), ζ∗t (x) = −Ṽy(Ux(x)), C∗t (c) = ζ∗t (X∗t (x)) (1.6)

(iv) By Equation (1.5), strong consistency of (U, V,X∗) implies the monotony of y 7→Y ∗t (y). The system (Ũ , Ṽ , Y

∗) is strongly Y -consistent.

The main consequence of the strong consistency is to provide a closed form for consump-

tion consistent utility system.

Theorem 1.6. Let ζ̄t(x) be a positive progressive process, increasing in x and let X̄t(x)

be a strictly monotonic solution with inverse X̄t(z) of the SDE,

dX̄t(x) = X̄t(x)[rtdt+ κ∗t (X̄t(x))(dWt + ηRt dt)]− ζ̄t(X̄t(x)) dt, κ∗t (X̄t(x)) ∈ Rt.

Let Ȳt(y) be a strictly monotonic solution with inverse (Ȳt(z)) of the SDE

dȲt(y) = Ȳt(y)[− rtdt+ (ν∗t (Ȳt(y))− ηRt ).dWt

], ν∗t (Ȳt(y)) ∈ R⊥t .

Given a deterministic utility system (u, v) such that ζ̄0(x) = ζ(x) = −ṽy(ux(x)), thereexists a X c-consistent progressive utility system (U, V ) such that (X̄t(x), Ȳt(y)) are the

associated optimal processes, de�ned by:

Ux(t, x) = Ȳt(ux(X̄t(x)), Vc(t, c) = Ux(t, ζ̄−1t (c)) with ζ̄−1(c) = u−1x (vc(c)) (1.7)

Observe that the consumption optimization contributes only through the conjugate Ṽ of

the progressive utility V. We refer to [16] for detailed proofs.

1.3 X c-consistent utilities with linear optimal processes

The simplest example of monotonic process is given by linear processes with positive

(negative) stochastic coe�cient. It is easy to characterize consumption consistent utility

sytems (U, V ) associated with linear optimal processes

X∗t (x) = xX∗t with X

∗t := X

∗t (1), Y

∗t (y) = yY

∗t with Y

∗t := Y

∗t (1)

6

Proposition 1.7. (i) A strongly X c-consistent progressive utility (U, V ) generates linear

optimal wealth and state price processes if and only if it is of the form

U(t, x) = Y ∗t X∗t u(

x

X∗t), V (t, c) = ζ∗t U(t,

c

ζ∗t) with ζt(x) = x ζt.

The optimal processes are then given by

X∗t (x) = xX∗t , Y

∗t (y) = yY

∗t , and C

∗t (x) = X

∗t (x)ζ

∗t .

(ii) Power utilities A consumption consistent progressive power utility (U θ, V θ) (with

risk aversion coe�cient θ) generates necessarily linear optimal processes and is, conse-

quently, of the form(U (θ)(t, x) = Y

∗t X∗t

1−θ(xX∗t

)1−θ, V (θ)(t, x) = (ψ̂t)θ

Y ∗t X∗t

1−θ(xX∗t

)1−θ).

Proof. If X∗t (x) = xX∗t and Y

∗t (y) = yY

∗t , their inverse �ows are also linear and X (t, x) =

xX∗t,Y(t, y) = yY ∗t .

(i) a) The linearity with respect to its initial condition of the solution of one dimen-

sional SDE with drift bt(x) and di�usion coe�cient σt(x) can be satis�ed only when the

coe�cients bt(x) and σt(x) are a�ne in x, that is bt(x) = xbt and σit(x) = xσit , b and

σi being one dimensional progressive processes. Since the only coe�cient with some non

linearity in the dynamics of Y ∗t (y) is yν∗t (y), the previous condition implies that ν

∗t (y)

does not depend on y. By the same argument, we see that xκ∗t (x) is linear and κ∗t (x) also

does not depend on x. For the consumption process, ζ∗t (x) the linear condition becomes

ζ∗t (x) = xζ∗t .

b) We are concerned by strongly consistent progressive utilities, since optimal processes

are monotonic by de�nition. Then, since Y ∗t (ux(x)) = Ux(t,X∗t (x)), we see that the

marginal utility Ux is given by Ux(t, x) = ux(x/X∗t )Y∗t . By taking the primitive with the

condition U(t, 0) = 0, U is given by U(t, x) = u(x/X∗t )X∗t Y∗t .

c) We know that C∗t (x) = X∗t (x)ζ

∗t and ζ̄t(t,X

∗t (x)) = −Ṽy(t, Ux(t,X∗t (x))) (from opti-

mality conditions). Thus −Ṽy(t, Ux(t,X∗t (x))) = X∗t (x)ζ∗t, a.s. ∀t, x. From monotonicityof X∗ and Ux, we then conclude that Vc(t, c) = Ux(t, cζ∗t ) a.s. ∀t, c. Integrating yields thedesired formula.

(ii) Power-type utilities generate linear optimal processes. So, we only have to con-

sider initial power utilities u(x) = kx1−θ/(1 − θ), v(c) = kvc1−θ/(1 − θ) with the samerisk-aversion coe�cient θ to characterize the system.

Remark 1.1. In order to separate the messages and as the risk aversion does not vary in

this result, we have deliberately omitted the indexing of the optimal process by θ, especially

in the explicit case of power utilities. Although, optimal process may re�ect a part of this

risk aversion, therefore in the last section of this work, we take care to make them also

dependent on this parameter

7

1.4 Value function of backward classical utility maximiza-

tion problem

As for example in the Ramsey rule, utility maximization problems in the economic liter-

ature use classical utility functions. This subsection points out the similarities and the

di�erences between consistent progressive utilities and backward classical value functions,

and their corresponding portfolio/consumption optimization problems.

Classical portfolio/consumption optimization problem and its conjugate

problem The classic problem of optimizing consumption and terminal wealth is de-

termined by a �xed time-horizon TH and two deterministic utility functions u(.) and

v(t, .) de�ned up to this horizon. Using the same notations as previously, the classical

optimization problem is formulated as the following maximization problem,

sup(κ,c)∈X c

E(u(Xκ,cTH ) +

∫ TH0

v(t, ct)dt). (1.8)

For any [0, TH ]-valued F-stopping τ and for any positive random variable Fτ -mesurableξτ , X c(τ, ξτ ) denotes the set of admissible strategies starting at time τ with an initial

positive wealth ξτ , stopped when the wealth process reaches 0. The corresponding value

system (that is a family of random variables indexed by (τ, ξτ )) is de�ned as,

U(τ, ξτ ) = ess sup(κ,c)∈X c(τ,ξτ ) E(u(Xκ,cTH (τ, ξτ )) +

∫ THτ

v(s, cs)ds|Fτ), a.s. (1.9)

with terminal condition U(TH , x) = u(x).We assume the existence of a progressive utility still denoted U(t, x) that aggregates thesesystem (that is more or less implicit in the literature). When the dynamic programming

principle holds true, the utility system (U(t, x), v(t, .) is X c-consistent. Nevertheless, inthe backward point of view, it is not easy to show the existence of optimal monotonic

processes, or equivalently the strong consistency. Besides, the optimal strategy in the

backward formulation is not myopic and depends on the time-horizon TH . In the economic

literature, TH is often taken equal to +∞ and the utility function is separable in timewith exponential decay at a rate β interpreted as the pure time preference parameter:

v(t, c) = e−βtv(c). It is implicitly assumed that such utility function are equal to zero

when t tends to in�nity.

2 Ramsey rule and Yield Curve Dynamics

As our aim is to study long term a�ne yields curves, we will focus in the following on

a�ne optimal processes. But let us �rst recall some results on the Ramsey rule with

progressive utility.

8

2.1 Ramsey rule

Financial market cannot give a satisfactory answer for the modeling of long term yield

curves, since for longer maturities, the bond market is highly illiquid and standard �nan-

cial interest rates models cannot be easily extended.

Economic point of view of Ramsey rule Nevertheless, an abundant literature

on the economic aspects of long-term policy-making has been developed. The Ramsey

rule is the reference equation in the macroeconomics literature for the computation of long

term discount factor. The Ramsey rule comes back to the seminal paper of Ramsey [26]

in 1928 where economic interest rates are linked with the marginal utility of the aggregate

consumption at the economic equilibrium. More precisely, the economy is represented by

the strategy of a risk-averse representative agent, whose utility function on consumption

rate at date t is the deterministic function v(t, c). Using an equilibrium point of view with

in�nite horizon, the Ramsey rule connects at time 0 the equilibrium rate for maturity T

with the marginal utility vc(t, c) of the random exogenous optimal consumption rate (Cet )

by

Re0(T ) = −1T

lnE[vc(T,CeT )]

vc(c). (2.1)

Remark that the Ramsey rule in the economic literature relies on a backward formulation

with in�nite horizon, an usual setting is to assume separable in time utility function

with exponential decay at rate β > 0 and constant risk aversion θ, (0 < θ < 1), that is

v(t, c) = Ke−βt c1−θ

1−θ . β is the pure time preference parameter, i.e. β quanti�es the agent

preference of immediate goods versus future ones. Ce is exogenous and is often modeled

as a geometric Brownian motion.

In the �nancial point of view we adopt here the agent may invest in a �nancial market

in addition to the money market. We consider an arbitrage approach with exogenously

given interest rate, instead of an equilibrium approach that determines them endogenously.

It seems also essential for such maturity to adopt a sequential decision scheme that allows

to revise the �rst decisions in the light of new knowledge and direct experiences: the

utility criterion must be adaptative and adjusted to the information �ow. That is why we

consider consistent progressive utility. The �nancial market is an incomplete Itô �nan-

cial market: notations are the one described in Section 1.1, with a n standard Brownian

motion W , a (exogenous) �nancial short term interest rate (rt) and a n-dimensional risk

premium (ηRt ). In the following, we adopt a �nancial point of view and consider either

the progressive or the backward formulation for the optimization problem.

Marginal utility of consumption and state price density process

(i) The forward dynamic utility problem

Proposition 1.5 gives a pathwise relation between the marginal utility of the optimal

consumption and the optimal state price density process, where the parameterization is

9

done through the initial wealth x, or equivalently c or y since c = −ṽy(ux(x)) = −ṽy(y),

Vc(t, C∗t (c)) = Y∗t (y), t ≥ 0 with vc(c) = y. (2.2)

The forward point of view emphasizes the key rule played by the monotony of Y with

respect to the initial condition y, under regularity conditions of the progressive utilities

(cf [15]). Then as function of y, c is decreasing, and C∗t (c) is an increasing function of

c. This question of monotony is frequently avoided, maybe because with power utility

functions (the example often used in the literature) Y ∗t (y) is linear in y as ν∗ does not

dependent on y. We shall come back to that issue in Section 4.

(ii) The backward classical optimization problem

In the classical optimization problem, both utility functions for terminal wealth and con-

sumption rate are deterministic, and a given horizon TH is �xed. In this backward point

of view, optimal processes are depending on the time horizon TH : in particular the opti-

mal consumption rate C∗,H(y) depends on the time horizon TH through the optimal state

price density process Y ∗,H , leading to the same pathwise relation (2.2) as in the forward

case,vc(t, C

∗,Ht (c))

vc(c)=Y ∗,Ht (y)

y, 0 ≤ t ≤ TH with vc(c) = y. (2.3)

So, in general the notation of the forward case are used, but with the additional symbol H

(Y ∗,H , c∗,H , X∗,H) to address the dependency on TH in the classical backward problem.

Conclusion: Thanks to the pathwise relation (2.2), the Ramsey rule yields to a de-

scription of the equilibrium interest rate as a function of the optimal state price density

process Y ∗,e, Re0(T )(y) = − 1T ln E[Y∗,eT (y)/y], that allows to give a �nancial interpretation

in terms of zero coupon bonds. More dynamically in time, ∀t < T,

Ret (T )(y) := −1

T − tln E

[Vc(T,C

∗,eT (c))

Vc(t, C∗,et (c))

∣∣Ft] = − 1T − t

ln E[Y ∗,eT (y)Y ∗,et (y)

∣∣Ft] . (2.4)2.2 Financial yield curve dynamics

Based on the foregoing, it is now proposed to make the connection between the economic

and the �nancial point of view through the state price densities processes and the pricing.

Let(Bm(t, T ), t ≤ T

), (m for market), be the market price at time t of a zero-coupon

bond paying one unit of cash at maturity T . Then, the market yield curve is de�ned as

usual by the actuarial relation, Bm(t, T ) = exp(−Rmt (T )(T − t)). Thus our aim is to givea �nancial interpretation of E

[Y ∗T (y)Y ∗t (y)

∣∣Ft] for t ≤ T in terms of price of zero-coupon bonds.Remark that Y ∗ is solution of an optimization problem whose criteria depend on the utility

functions, yet the utilities do not intervene in the dynamics of Y ∗. In term of pricing, the

terminal wealth at maturity T represents the payo� at maturity of the �nancial product,

whereas the consumption may be interpreted as the dividend distributed by the �nancial

product before T .

10

Replicable bond For admissible portfolio without consumption Xκt , it is straightfor-

ward that for any state price process Y νt Xκt is a local martingale, and so under additional

integrability assumption, Xκt = E[XκT

Y νTY νt

∣∣Ft]. So the price of XκT does not depend on ν.This property holds true for any derivative whose the terminal value is replicable by an

admissible portfolio without consumption, for example a replicable bond,

Bm(t, T ) = E[Y 0TY 0t

∣∣Ft] = EQ[e− R Tt rsds∣∣Ft] = E[YT (y)Yt(y)

∣∣Ft]Besides, Bm(t, T ) = E

[YT (y)

Yt(y)

∣∣Ft] for any state price density process Y with goods inte-grability property.

Non hedgeable bond For non hedgeable zero-coupon bond, the pricing by indi�er-

ence is a way (among others) to evaluate the risk coming from the unhedgeable part.

The utility indi�erence price is the cash amount for which the investor is indi�erent be-

tween selling (or buying) a given quantity of the claim or not. This pricing rule is non

linear and provides a bid-ask spread. If the investor is aware of its sensitivity to the

unhedgeable risk, they can try to transact for a little amount. In this case, the �fair price�

is the marginal utility indi�erence price (also called Davis price [2]), it corresponds to the

zero marginal rate of substitution. We denote by Bu(t, T ) (u for utility) the marginal

utility price at time t of a zero-coupon bond paying one cash unit at maturity T , that

is Bu(t, T ) = But (T, y) = E[Y ∗T (y)Y ∗t (y)

∣∣Ft]. Based on the link between optimal state pricedensity and optimal consumption, we see that

But (T, y) := Bu(t, T )(y) = E

[Y ∗T (y)Y ∗t (y)

∣∣Ft] = E[Vc(T,C∗T (c))Vc(t, C∗t (c))

∣∣Ft], vc(c) = y. (2.5)Remark that Bu(t, T )(y) is also equal to E

[Ux(T,X∗T (x))Ux(t,X∗t (x))

∣∣Ft]. Nevertheless, besides the eco-nomic interpretation, the formulation through the optimal consumption is more relevant

than the formulation through the optimal wealth : indeed the utility from consumption

V is given, while the utility from wealth U is more constrained.

According to the Ramsey rule (2.4), equilibrium interest rates and marginal utility inter-

est rates are the same. Nevertheless, for marginal utility price, this last curve is robust

only for small trades.

The martingale property of Y ∗t (y)But (T, y) yields to the following dynamics for the zero

coupon bond maturing at time T with volatility vector Γt(T, y)

dBut (T, y)But (T, y)

= rtdt+ Γt(T, y).(dWt + (ηRt − ν∗t (y))dt). (2.6)

Using the classical notation for exponential martingale, Et(θ) = exp( ∫ t

0 θs.dWs−12

∫ t0 ‖θs‖

2.ds),

the martingale Y ∗t (y)But (T, y) can written as an exponential martingale with volatility(

ν∗. (y)−ηR. +Γ.(T, y)). Using that BuT (T, y) = 1, we have two characterisations of Y

∗T (y),

Y ∗T (y) = Bu0 (T, y)ET

(ν∗. (y)− ηR. + Γ.(T, y)

)= y e−

R T0 rsdsET

(ν∗. (y)− ηR.

).

11

Taking the logarithm gives∫ T0rsds = TRu0(T )−

∫ T0

Γt(T, y).(dWt + (ηt − ν∗t (y))dt) +12‖Γt(T, y)‖2dt. (2.7)

2.3 Yield curve for in�nite maturity and progressive utilities

The computation of the marginal utility price of zero coupon bond is then straightforward

using (2.5) leading to the yield curve dynamics (Rut (T, y) = − 1T−t lnBut (T, y))

Rut (T, y) =T

T − tRu0(T, y)−

1T − t

∫ t0rsds−

∫ t0

Γs(T, y)T − t

dWs

+∫ t

0

||Γs(T, y)||2

2(T − t)ds+

∫ t0<

Γs(T, y)T − t

, ν∗s − ηRs > ds.

for �nite maturity, and lut (y) := limT→+∞Rut (T, y) for in�nite maturity.

As showed in Dybvig [24] and in El Karoui and alii. [23] the long maturity rate lut (y)

behaves di�erently according to the long term behavior of the volatility when T →∞,

− If limT→∞ Γt(T,y)T−t 6= 0, a.s., then||Γt(T,y)||2

T−t →∞ a.s and lt(y) is in�nite.

− Otherwise, lt = l0 +∫ t

0 limT→∞(||Γs(T,y)||2

2(T−s)

)ds, and lt is a non decreasing process,

constant in t and ω if limT→∞||Γt(T,y)||2

T−t = 0.

In this last case, which is the situation considered by the economists, all past, present or

future yield curves have the same asymptote.

3 Progressive utilities and yield curves in a�ne

factor model

Recently, a�ne factor models have been intensively developed with some success to cap-

ture under the physical probability measure both �nancial and macroeconomics e�ects,

from the seminal paper of Ang and Piazzesi (2003). As explained in Bolder&Liu (2007)

[1], A�ne term-structure models have a number of theoretical and practical advantages.

One of the principal advantages is the explicit description of market participants aggregate

attitude towards risk. This concept, captured by the market price of risk in particular, pro-

vides a clean and intuitive way to understand deviations from the expectations hypothesis

and simultaneously ensure the absence of arbitrage.

3.1 De�nition of a�ne market

The a�ne factor model makes it possible to compute tractable pricing formulas, it extends

the log-normal model (studied in [16]) to a more stochastic model. A�ne model, which

generalizes the CIR one, was �rst introduced by D. Du�e and R. Kan (1996) [3], where

the authors assume that the yields are a�ne function of stochastic factors, which implies

12

an a�ne structure of the factors. Among many others, M. Piazzesi reports in [25] some

recent successes in the study of a�ne term structure models. Several constraints must be

ful�lled to de�ne an a�ne model in mutidimentional framework, but we will not discuss

the details here and refer to the works of Teichmann and coauthors [17], [18].

Properties of a�ne processes and their exponential We adopt the framework

of the example in Piazzesi ([25], p 704). The factor is a N-dimensionnal vector process

denoted by ξ and is assumed to be an a�ne di�usion process, that is the drift coe�cient

and the variance-covariance matrix are a�ne function of ξ :

dξt = δt(ξt)dt+ σt(ξt)dWt (3.1)

The a�ne constraint is expressed as:

- δt(ξt) = %δt ξt + δ0t , where %

δt ∈ RN×N and δ0t ∈ RN are deterministic.

- σt(ξt) = Θtst(ξt), where Θt ∈ RN×N is deterministic, and the matrix st(ξt) is adiagonal N ×N matrix, with eigenvalues sii,t(ξt). The a�ne property concerns thevariance covariance matrix Θtst(ξt)s̃t(ξt)Θ̃t or equivalently (since st(ξt) is diagonal)

the positive eigenvalues λi,t = s2ii,t(ξt) of st(ξt)s̃t(ξt): λi,t = ρ̃λi,tξt+λ

0i,t that must be

positive 1 with deterministic (λ0i,t, ρ̃λi,t) ∈ R×RN , where .̃ denotes the transposition

of a vector or a matrix.

Characterization of market with a�ne optimal processes To be coherent

with the previous market model (Section 1.1), we have to de�ne the set of admissible

strategies Rt , at date t, and its orthogonal R⊥t . Let us �rst to point out that the (N ×1)volatility vector of any process ãtξt (a is deterministic) is given by st(ξt)Θ̃t.at. Thus if R(linear space) is the set of admissible strategies, then at time t it depends on ξt and is

necessarily given by,

Rt(ξt) = {ãtΘtst(ξt), at progressive vector in some linear and deterministic space Et ⊂ RN}

The deterministic space Et and its orthogonal are assumed to be stable by Θ̃t, or equiva-

lently the matrix Θ̃t is commutative with the orthogonal projection on Et. A block matrix

Θt (up to an orthogonal transformation) satis�es this property. Furthermore, Et and E⊥tare stable by %δt . The set of admissibles strategies being well de�ned, we denote by a

Rt

and by a⊥t the elements of the linear spaces Et and E⊥t .

We consider two types of assumptions:

(i) The spot rate (rt) and the consumption rate (ζt) are a�ne positive processes

rt = art ξt + brt and ζt = a

ζt ξt + b

ζt .

(ii) The volatilities of the optimal processes X∗ and Y ∗ have a�ne structure,

κ∗t = ãX,Rt Θtst(ξ), η

Rt = ã

Y,Rt Θtst(ξ), ν

∗t = ã

Y,⊥t Θtst(ξ), (a

.,R ∈ Et and a.,⊥ ∈ E⊥t ).(3.2)

1This constraint is restrictive, but we do not go into the details of this hypothesis as this is not the goal of

this work, for more details see the literature cited above concerning a�ne models.

13

Forward utility and marginal utility yields curve In the forward utility frame-

work with linear portfolios, the marginal utility price of zero-coupon bond with maturity

T is given by Equation (2.5), where Y ∗t (y) is linear in y. The price of zero-coupon does

not depend on y. Taking into account the speci�cities of the a�ne market, we have that

But (T ) = E[Y ∗TY ∗t

∣∣Ft] (3.3)= E

[exp

(−∫ Tt

(arsξs + brs)ds+

∫ TtãYu Θusu(ξ)dWu −

12

∫ Tt‖ãYu Θusu(ξ)‖2du

)∣∣Ft]Thanks to the Markovian structure of the a�ne di�usion ξ, the price of zero-coupon bond

is an exponential a�ne fonction of ξ, with the terminal constraint BuT (T ) = 1,

ln(But (T )) = ÃTt ξt +B

Tt , A

TT = 0, B

TT = 0.

Ricatti equations To justify the Ricatti equations, we �x some notations and develop

useful calculation for study exponential a�ne process. Moreover, when working with

a�ne function f(t, ξ) = ãtξ + bt, it is sometimes useful to write bt = ft(0) = f0t and

at = ∇ξft(0) = ∇f0t .

Lemma 3.1. Let at ∈ RN and bt ∈ R be two deterministic functions.(i) The a�ne process ãtξt + bt is a semimartingale with decomposition,

d(ãtξt + bt) = ãtΘtst(ξ)dWt −12||ãtΘtst(ξ)||2dt+ ft(at, ξt)dt

a) The quadratic variation ||ãtΘtst(ξ)||2 = qt(a, ξt) is quadratic in a and a�ne in ξ. Moreprecisely, if Θit is the i-th column of the matrix Θ, we have

qt(a, ξ) = ∇q0t (a)ξ + q0t (a), with∇q0t (a) = Σi=Ni=1(Θ̃itat

)2ρλi,t, and q

0t (a) = Σ

i=Ni=1

(Θ̃itat

)2λ0i,t

b) The drift term ft(a, ξt) is an a�ne quadratic form in a and a�ne in ξ given by

ft(a, ξt) = (∂tãt + ãt%δt +12∇q0t (a)ξt + ∂tbt + ãtδ0t +

12q0t (a) (3.4)

(ii) A process Xt = ãtξt + bt +∫ t

0 δXs (ξs)ds (δ

Xt (ξ) = ∇δ

0,Xt ξ + δ

0,Xt ) is the log of an

exponential martingale if and only if the coe�cients satisfy the Ricatti equation

∂tãt + ãt%δt +12∇q0t (a) +∇δ

0,Xt = 0, ∂tbt + ãtδ

0t +

12q0t (a) + δ

0,Xt = 0 (3.5)

Proof. (i) From Itô's formula, d(ãtξt + bt) = ∂t(ãt)ξtdt+ ãtdξt + ∂tbtdt,

which implies from the dynamics (3.1) of ξ,

d(ãtξt + bt) =(∂t(ãt)ξt + ãtδt(ξt) + ∂tbt

)dt+ ãtΘtst(ξ)dWt

=(∂t(ãt)ξt + ãtδt(ξt) + ∂tbt +

12||ãtΘtst(ξ)||2

)dt+

(ãtΘtst(ξ)dWt −

12||ãtΘtst(ξ)||2dt

).

a) The sequel is based on the decomposition of the quadratic variation of ãtξt+bt as a�ne

form in ξt and quadratic in a.

b) Then, the a�ne decomposition of the drift term may be rewritten in the same way.

14

(ii) exp(Xt) will be an exponential martingale, if and only if dXt = ãtΘtst(ξ)dWt −12 ||ãtΘtst(ξ)||

2dt, or equivalently if and only if ft(a, ξ)+δX(t, ξ) ≡ 0. Given that ft(a, ξ)+δX(t, ξ) is a�ne in ξ, the condition is satis�ed if both, the coe�cient of ξ and the constant

term are null functions.

Application to bond pricing Solving Ricatti Equation (3.5) with any given initial

condition ã0ξ0 + b0 implies that the exponential a�ne process exp(ãtξt+ bt+∫ t

0 δXs (ξs)ds)

is a local martingale.

In a backward formulation, solving Ricatti Equation (3.5) with terminal constraint XT =

ãT ξT+bT+∫ T

0 δXs (ξs)ds implies under additional integrability assumption, that, if (a

Tt , b

Tt )

are solutions of the Riccatti system with terminal condition (aT , bT ).

E[

exp(XT −Xt)|Ft]

= exp(ãTt ξt + b

Tt

). (3.6)

In the theory of bond pricing, from Equation (3.3), we are looking for an a�ne process

ÃTt ξt +BTt such that, since rt = a

rt ξt + b

rt ,

exp(ÃTt ξt+B

Tt

)= E

[exp

( ∫ Tt−(art ξt+brt )du+ãYu Θusu(ξ)dWu−

12‖ãYu Θusu(ξ)‖2du

)|Ft]

Given that the martingale part is the one associated with the a�ne process ãYt ξt, it is

possible to de�ne an a�ne process XY with a�ne drift such that,

XYt = ãYt ξt −

∫ t0fs(aYs , ξs) ds = X

Y0 +

∫ t0ãYu Θusu(ξ)dWu −

12‖ãYu Θusu(ξ)‖2du (3.7)

where ft(a, ξ) is de�ned in Equation (3.4) with b = 0. Since exp(XYt ) is a local martingale,

ãYt is solution of the Ricatti equation with bt ≡ 0. In the new formulation, we arelooking for some process XTt = Ã

Tt ξt + B

Tt such that the exponential of the process

Zt = XTt + XYt −

∫ t0 rsds is a local martingale. The problem belongs to the family of

previous problem applied to the process Z. We summarize these results below.

Theorem 3.2. Assume an a�ne optimization framework, where the optimal state price

has an a�ne volatility ãYt Θtst(ξ) with ãYt solution of some Ricatti equation.

(i) Any zero-coupon bond is an exponential function exp(ÃTt ξt + B

Tt

)such that ãZt =

ÃTt + ãYt is solution of a Ricatti function with terminal condition ã

YT , and δ

Z function

δZt (ξ) = −ft(aYt ξ) + art ξt + brt .(ii) The volatility of a bond with maturity T is Γt(T ) = ÃTt Θtst(ξ).

Proof. The process Zt = XTt +XYt −

∫ t0 rsds is an a�ne process with a�ne integral term.

The decomposition of Z is of the type Zt = ãZt ξt + bZt +

∫ t0 δ

Zs (ξs)ds with ã

Zt = Ã

Tt + ã

Yt ,

bZt = BTt , δ

Zt (ξ) = −ft(aYt , ξ) + art ξt + brt . So ãZt satis�es a Ricatti equation with terminal

value ãYT .

15

3.2 A�ne model and power utilities

As power utilities is the classical most important example for economics, we now study

the marginal utility yield curve in a�ne model with progressive and backward power util-

ities. The backward case di�ers signi�cantly of the forward case, since constraints appear

on the optimal processes at maturity TH .

Backward formulation In the classical backward problem with classical power util-

ity function xθ, (0 < θ < 1) and horizon TH , we have the terminal constraint on the

optimal processes: the terminal values of the optimal wealth process X∗,HTH and of the

state price process Y ∗,HTH satisfy (from optimality)

(Y ∗,HTH )1/θX∗,HTH = Cst = exp k

We recall here by the index H the dependency on the horizon TH ; moreover for simplicity

we make k = 0.

The question is then how this constraint is propagated at any time in an a�ne frame-

work, with a�ne consumption rate ζ∗t . For notational simplicity, we denote X∗,ζ,H (the

optimal wealth process capitalized at rate ζ∗t by the process S0,ζ∗

t = exp∫ t

0 ζ∗udu.

Using that S0,ζ∗X∗,HY ∗,H is a martingale with terminal value S0,ζ

∗

TH(Y ∗,HTH )

1−1/θ, we study

the martingale M θt = E(S0,ζ

∗

TH(Y ∗,HTH )

1−1/θ|Ft)in two ways:

(i) the �rst one is based on the process X∗,Ht , since Mθt = S

0,ζ∗

t X∗,Ht Y

∗,Ht

(ii) the second one is very similar to the study of zero-coupon bond, by observing that by

the Markov property E[(S0,ζ

∗

TH/S0,ζ

∗

t )(Y∗,HTH

/Y ∗,Ht )1−1/θ|Ft

]is an exponential a�ne, whose

coe�cients (Aθ, Bθ) are solutions of a Ricatti equation, that is

E((S0,ζ

∗

TH/S0,ζ

∗

t )(Y∗,HTH

/Y ∗,Ht )1−1/θ|Ft

)= exp(Ãθt ξt +B

θt )

The backward constraint is then equivalent to the stochastic equality

S0,ζ∗

t X∗,Ht Y

∗,Ht = S

0,ζ∗

t (Y∗,Ht )

1−1/θ exp(Ãθt ξt +Bθt ).

Proposition 3.3. (i) The terminal optimal constraint X∗,HTH = S0,ζ∗

TH(Y ∗,HTH )

−1/θ is prop-

agated through the time into a closed relation with the state price process,

X∗,Ht = (Y∗,Ht )

−1/θ exp(Ãθt ξt +Bθt ), (3.8)

where exp(Ãθt ξt +Bθt ) = E

[(S0,ζ

∗

TH/S0,ζ

∗

t )(Y∗,HTH

/Y ∗,Ht )1−1/θ|Ft

].

(ii) In particular, the constraint that the volatility κ∗t has an a�ne structure belonging to

the space E, implies that (Ãθt )⊥ = 1/θ(ãYt )

⊥.

(iii) The links with the zero-coupon bond is given by the relation,

B(t, T ) = exp(ÃTt ξt +BTt ) = E

(Y ∗,HTY ∗,Ht

|Ft). (3.9)

16

4 Yield curve dynamics non-linear on initial condi-

tions

4.1 From linear optimal processes to more general progres-

sive utilities

Until then, we have omitted the dependence of optimal processes with respect to risk

aversion. In what follows, risk aversion plays an important role, therefore, an agent that

has as risk aversion denoted θ, his utility process will be denoted U θ, his optimal wealth

is denoted X∗θ and �nally his optimal dual process Y ∗,θ. For simplicity we are concerned

in the following only by utility processes which are of power type. As we have already

mentioned above, utilities of power type generate optimal processes X∗,θ and Y ∗,θ which

are linear with respect to their initial conditions, i.e, X∗,θ(x) = xX∗,θ and Y ∗,θ(y) = yY ∗,θ

(with X∗,θ = X∗,θ(1) and Y ∗,θ = Y ∗,θ(1)). Thus the marginal utility price at time t of a

zero coupon with maturity T , given in (2.5) by

Bθ(t, T )(y) := Bu(t, T )(y) = E[Y ∗,θT (y)Y ∗,θt (y)

∣∣Ft] = E[Y ∗,θt,T ∣∣Ft], with Y ∗,θt,t = 1does not depend on y nor on the consumption of the market at time t. This is not sur-

prising given that power utilities, although they are useful to compute explicit optimal

strategies, are somehow restrictive because they generates only linear optimal processes.

Besides, the economic litterature emphasizes the dependence of the equilibrium rate Re0(T )

on the initial consumption. To study this dependence, we have to give a nontrivial ex-

ample of stochastic utility that generates a nonlinear state price density process and then

calculate the price of zero coupon. This is not obvious, especially since our goal is to give

an explicit formula for the optimal dual process. The idea is to �rst generate, from opti-

mal process X∗,θ and Y ∗,θ associated with progressive power utilities U θ, a new processes

X̄ and Ȳ which are both admissible, monotone and especially nonlinear with respect to

their initial conditions. In a second step, we use the characterization (1.7) of Theorem 1.6

to thereby construct non-trivial stochastic utilities with X̄ and Ȳ as optimal processes.

The method that we will develop in the following is the starting point of the work of El

Karoui and Mrad [13], in which many other ideas and extensions can then be found.

step 1: To �x the idea, (X∗,θ, Y ∗,θ, ζ∗,θ) denotes the triplet of optimal primal, dual and

consumption processes associated with stochastic utility (U θ, V θ) of power type and rel-

ative risk aversion U θx/xUθxx = θ. We consider also two strictly decreasing probabil-

ity density functions f and g,∫ +∞

0 f(θ′)dθ′ =

∫ +∞0 g(θ

′)dθ′ = 1, use the change of

variable θ = zθ′ and de�ne the strictly increasing functions xθ(x) := f(θ/x), x > 0,

yθ(y) := g(θ/y), y > 0 satisfying the following identities∫ +∞0

xθ(x)dθ = x, ∀x > 0

17

∫ +∞0

yθ(y)dθ = y, ∀y > 0

step 2: We are now concerned with the following processes X̄, Ȳ and ζ̄ de�ned by

X̄t(x) :=∫ +∞

0 xθ(x)X∗,θt dθ, x > 0 (4.1)

Ȳt(y) :=∫ +∞

0 yθ(y)Y ∗,θt dθ, y > 0 (4.2)

where we recall that X∗,θt (resp. Y∗,θt ) denotes, in this integral, the optimal process

starting from the initial condition equal to 1. As seen previously, these two processes are

an admissible wealth and a state density process which are strictly increasing with respect

to their initial conditions of which they depend on non-trivial way far from being linear.

The consumption ζ̄ intuitively associated with X̄ is given by

ζ̄t(X̄t(x)) =∫ +∞

0ζ∗,θt (X

∗,θt (x

θ(x)))dθ =∫ +∞

0xθ(x)ζ∗,θt (X

∗,θt )dθ (4.3)

where the last equality comes from the linearity, for a �xed θ, of ζ∗,θ andX∗,θ. To complete

the construction of the progressive utility for which (X̄, Ȳ , ζ̄) will be the optimal processes,

a martingale property on the process (e−R t0 ζ̄sdsX̄tȲt) is necessary. We, then, make the

following assumption

Assumption 4.1. The optimal policies κ∗,θ and ν∗,θ and the market risk premium η are

uniformly bounded.

Assumption 4.1 implies that (e−R .0 ζ∗,θs dsX∗,θY ∗,θ

′) are martingales for all θ, θ′ and conse-

quently (e−R t0 ζ̄sdsX̄tȲt) is also a martingale.

step 3: The last step is then to consider any classical utility functions u and v (not nec-

essarily of power type nor generating linear optimal processes) and only impose that their

derivatives ux and vc have good integrability conditions close to zero. All the ingredients

were met, from (1.7) of Theorem 1.6, by considering the monotonic process C̄ de�ned by

C̄(v−1c (ux(x))) := ζ̄t(X̄t(x)) =∫ +∞

0ζ∗,θt (X

∗,θt (x

θ(x)))dθ,

the pair of random �elds de�ned by{U(t, x) =

∫ x0 Ȳt(ux(X̄ (t, z))dz,

V (t, c) =∫ c

0 Ȳt(vc(C̄t(θ))dθ(4.4)

is a consistent progressive utility of investment and consumption generating (X̄, Ȳ , C̄) as

optimal wealth, dual and consumption processes, with dual (Ũ , Ṽ ):{Ũ(t, y) =

∫∞y X̄t(−ũy(Ȳ(t, z))dz,

Ṽ (t, c) =∫∞c C̄t(−ṽc(Ȳ(t, α))dα

(4.5)

where (X̄ , Ȳ, C̄) denotes the inverse �ows of X̄, Ȳ and C̄.Example Suppose that for any θ, θ′ we have X∗,θ = X∗,θ

′= X∗, ∀θ a.s., then in this

18

case X̄(x) = xX∗ with inverse X̄ (x) = x/X∗, consequently the progressive utility U isgiven by:

U(t, x) =∫ x

0

∫ +∞0

yθ(ux(x/X∗t ))Y∗,θt dθdz.

4.2 Application to Ramsey rule evaluation

Let us now, turn to the Ramsey rule, and study the price of zero coupon. We recall at

�rst that the price in our new framework is then given by

B(t, T )(y) = E[ ȲT (y)Ȳt(y)

∣∣Ft] (4.6)From the formula of Ȳ (4.2), the price B(t, T ) becomes

B(t, T )(y) =1∫ +∞

0 yθ(y)Y ∗,θt dθ

E[ ∫ +∞

0yθ(y)Y ∗,θT dθ

∣∣Ft] (4.7)Now, let us introduce Bθ(t, T ) the zero coupon bond (independent on y because Y ∗,θ is

linear on y) associated with risk aversion θ de�ned by

Bθ(t, T ) = E[Y ∗,θT (y)Y ∗,θt (y)

∣∣Ft] (4.8)it follows that

B(t, T )(y) =1∫ +∞

0 yθ(y)Y ∗,θt dθ

∫ +∞0

yθ(y)Y ∗,θt Bθ(t, T )dθ (4.9)

It is clear from this formula, that the zero coupon is a mixture of prices Bθ(t, T ) weighted

by yθ(y)Y ∗,θtR +∞

0 yθ(y)Y ∗,θt dθ

which is strongly dependent on y of non-trivial way.

At this level, several questions naturally arise: What is the sensitivity of the bond with

respect to y? It is monotone, concave, convex? What about its asymptotic behavior?

Give complete and satisfactory answers to these questions is beyond the scope of this

work but will be addressed in a future paper.

References

[1] David Jamieson Bolder and Shudan Liu. Examining simple joint macroeconomic and

term-structure models: A practitioner's perspective. 2007.

[2] Mark H.A. Davis. Option pricing in incomplete markets. In S.R. Pliska, editor,

Mathematics of Derivative Securities, pages 216�226. M.A.H. Dempster and S.R.

Pliska, cambridge university press edition, 1998.

[3] D.Du�e and L.G.Epstein. Stochastic di�erential utility. Econometrica, 60(2):353�

394, 1992. With an appendix by the authors and C. Skiadas.

19

[4] Christian Gollier. What is the socially e�cient level of the long-term discount rate?

[5] Christian Gollier. An evaluation of sten's report on the economics of climate change.

Technical Report 464, IDEI Working Paper, 2006.

[6] Christian Gollier. Comment intégrer le risque dans le calcul économique? Revue

d'économie politique, 117(2):209�223, 2007.

[7] Christian Gollier. The consumption-based determinants of the term structure of

discount rates. Mathematics and Financial Economics, 1(2):81�101, July 2007.

[8] Christian Gollier. Managing long-term risks. 2008.

[9] Christian Gollier. Ecological discounting. IDEI Working Papers 524, Institut

d'Économie Industrielle (IDEI), Toulouse, July 2009.

[10] Christian Gollier. Expected net present value, expected net future value and the

ramsey rule. Technical Report 557, IDEI Working Paper, June 2009.

[11] Christian Gollier. Should we discount the far-distant future at its lowest possible

rate? Economics: the Open Access, Open-Assessment E-Journal, 3(2009-25), June

2009.

[12] I.Karatzas and S.E.Shreve. Methods of Mathematical Finance. Springer, September

2001.

[13] N.El Karoui and M. Mrad. Mixture of consistent stochastic utilities, and a priori

randomness. preprint., 2010.

[14] N.El Karoui and M. Mrad. Stochastic utilities with a given optimal portfolio : ap-

proach by stochastic �ows. Preprint., 2010.

[15] N.El Karoui and M. Mrad. An exact connection between two solvable sdes and a non

linear utility stochastic pdes. SIAM Journal on Financial Mathematics,, 4(1):697�

736, 2013.

[16] N.El Karoui, M. Mrad, and C. Hillairet. Ramsey rule with progressive utility

in long term yield curves modeling. preprint., 2014.

[17] M. Keller-Ressel, W. Schachermayer, and J. Teichmann. A�ne processes are regular.

Probability Theory and Related Fields, (151):591�611, 2011.

[18] M. Keller-Ressel, W. Schachermayer, and J. Teichmann. Regularity of a�ne processes

on general state spaces. Electronic Journal of Probability, 43(18):1�17, 2013.

[19] M.Musiela and T.Zariphopoulou. Stochastic partial di�erential equations in portfolio

choice. Preliminary report, 2007.

20

[20] M.Musiela and T.Zariphopoulou. Portfolio choice under dynamic investment perfor-

mance criteria. Quantitative Finance, 9(2):161�170, 2009.

[21] M. Musiela and T. Zariphopoulou. Backward and forward utilities and the associ-

ated pricing systems: The case study of the binomial model. pages 3�44. Princeton

University Press, 2005-2009.

[22] M. Musiela and T. Zariphopoulou. Investment and valuation under backward and

forward dynamic exponential utilities in a stochastic factor model. In Advances in

mathematical �nance, pages 303�334. Birkhäuser Boston, 2007.

[23] Antoine Frachot Nicole ElKaroui and Helyette Geman. On the behavior of long zero

coupon rates in a no arbitrage framework. Review of derivatives research, 1:351�369,

1997.

[24] J.E. Ingersol P.H. Dybvig and S.A. Ross. Long�orward and zero-coupon rates can

never fall. Journal of Business, 69:1�25, 1996.

[25] Monika Piazzesi. Handbook of Financial Econometrics, chapter A�ne Term Structure

Models, pages 691�766. 2010.

[26] F.P. Ramsey. A mathematical theory of savings. The Economic Journal, (38):543�

559, 1928.

[27] L.C.G. Rogers. A mathematical theory of savingsduality in constrained optimal

investment and consumption problems: A synthesis. Working paper, Statistical Lab-

oratory, Cambridge University., 2003.

[28] Martin L. Weitzman. Why the far-distant future should be discounted at its lowest

possible rate. Journal of Environmental Economics and Management, 36(3):201�208,

November 1998.

[29] Martin L. Weitzman. A review of the the stern review on the economics of climate

change. Journal of Economic Litterature, 45:703�724, September 2007.

21

Progressive Utility and Investment UniverseThe investment universeXc-consistent Utility and Portfolio optimization with consumptionXc-consistent utilities with linear optimal processes Value function of backward classical utility maximization problem

Ramsey rule and Yield Curve DynamicsRamsey rule Financial yield curve dynamicsYield curve for infinite maturity and progressive utilities

Progressive utilities and yield curves in affine factor modelDefinition of affine marketAffine model and power utilities

Yield curve dynamics non-linear on initial conditions From linear optimal processes to more general progressive utilities Application to Ramsey rule evaluation